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Example 6: Three phases, eutectic-type univariant line

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Consider three phases A, B and C in a binary system. The phase rule tells us that if A, B, and C all coexist in a binary system then the variance is one, so this should only be possible along a line or curve in P-T space. Let's begin at a random (P,T) point off this line and draw the -X diagram at this pressure and temperature. We might find a configuration like that in part (a) of the figure: the curve for ^C(X) is everywhere above the envelope of stability defined by the curves for A and B and their common internal tangent segment. Hence C is not stable at any bulk composition at this P and T; its existence has not changed the stability fields at all. Note that we can still find common tangents (shown in yellow) between A and C or between B and C, but these phase pairs are metastable because they always have higher than the A+B mixture.

Now let us imagine that phase C has a higher specific entropy than phases A or B (perhaps C is a liquid, for example). We showed above that the temperature derivative of at constant pressure is -S. So if we raise the temperature and draw a new -X plot, the curve for phase C will have moved down relative to the others (they all move down in absolute terms, but C moves down the most). As we keep raising the temperature at this pressure we will eventually reach a particular, special temperature at which the diagram looks like part (b) of the Figure. Here phases A, B, and C all share a common tangent. It is easy for two curves to have a common tangent, but it requires a special geometry to get three curves to share a tangent. This is a univariant condition, and between the ends of the green tangent segment at X_A(B) and X_B(A), we can have any combination of A+B, A+C, A+B, or A+B+C without any difference in Gibbs free energy.

Now let us keep increasing the temperature at constant pressure and draw a new -X diagram. The curve for phase C keeps moving down relative to the other two, and now there is no longer a common tangent (Part (c) of the figure). Furthermore, the A-B internal tangent segment is now metastable with respect to the A+C and C+B internal tangent segments and part of the C one-phase region. It is no longer possible to have all three phases together, because we have overstepped the univariant reaction. If we wanted to change the temperature and keep on the univariant, we would also have had to change the pressure, by just the right amount to bring the three phases back onto common tangent. In so doing we could map out a curve in P-T space where the univariant assemblage is stable, a univariant curve. The slope dP/dT we would have to follow to stay on the univariant curve is given by the Clausius-Clapeyron Equation and depends on the entropy and volume differences among the phases.

We can also see that crossing a univariant curve (i.e. going from the configuration in (a) to that in (c) by way of configuration (b)) causes a change in the sequence of stable phases across the X-axis. Thus a univariant curve is also a univariant reaction, in this case A+B=C. This means that on one side of the curve A+B is stable together, on the other side C is stable but A+B is not, and only on the curve can all three phases coexist.

If we take the sequences of stable phases along the X-axes of the three plots in the above figure and transfer them to an isobaric T-X diagram, filling in the spaces between with the sequences we would see if we moved continuously in temperature, we get the T-X diagram at right. The boundaries between the one-phase regions (colored black and labelled in red with the name of the stable phase) and the two-phase regions (colored white and labelled in green with the two coexisting phases) are, at the three temperatures corresponding to the three parts of the above figure, shown at the same X values as the grey lines we marked previously using common tangets. The univariant reaction is shown as a horizontal red line that touches the three one-phase regions and delineates the boundary between the stable two-phase regions.

Now examine the applet below. It has been initialized to example 6, which contains three phases A, B, and Liquid. There are three lines on the P-T panel. They represent the degenerate equilibria A=Liquid and B=Liquid (blue) and the univariant reaction A+B=Liquid (red). Try clicking in sequence three points at equal pressure (same X coordinate) that are below, on, and above the univariant as you change temperature. The changes in the -X panel should correspond exactly to what we just discussed. Now look at the T-X section at lower left...it should remind you of the T-X section we generated in the discussion above. The P-X diagram is similar but upside-down, since the stability field of liquid lies at high temperature but at low pressure.

Since this applet was written, browers have stopped supporting applets for security reasons. You can download a standalone executable that runs all the functionality of the applet on your own platform here. When you run it, select "Example 6" to get this page's content.